Logistic regression - specific hypothesis I am doing a logistic model explaining if the physical health of a person has been impacted or not during COVID. To do that, I am using many socio-demographics factors as sex, income, education level, and ethnicity. My hypothesis is that women with low income, less education, and are non-white are more impacted than other groups. How can I test this specific hypothesis ?
 A: Well, I think the outcome here is a bit wonky. I'm not sure how you operationalized  “greatly impacted” vs “less impacted”, but it seems that this could be defined any number of ways. Regardless, you have modeled it as a binary response, so this would easily be fit into a logistic regression. It also appears you feel there is a conditional relationship between your factors here. You could theoretically fit a four-way interaction between your variables like so:
$$
\text{COVID Impact} = \text{Gender}*\text{Race}*\text{Income}*\text{Education}
$$
To me this is a very hard to interpret model and honestly ignores the fact that income and education are both just proxies for overall socio-economic status (SES) anyway, so I would just combine them in some way (see Ware, 2017.)
Since you also seem to think there is a relationship between race and gender in this relationship, you could combine them as one categorical predictor (i.e. "Low-Income African-American, "Low-Income Caucasian", etc.). Then your model could potentially be much simpler to model as a two-way interaction:
$$
\text{COVID Impact} = \text{Gender-Race Intersectionality} * \text{SES}
$$
However, that may still complicate matters in it's own way by making several coefficients that don't directly test the effects of each group individually. So maybe the best balancing act you could model is this:
$$
\text{COVID Impact} = (\text{Gender} + \text{Race}) * \text{SES}
$$
This would allow you to test the main effects of $Gender$ and $Race$, as well as the interactions between $Gender * SES$ and $Race * SES$.
A: I'll simulate some data similar to your problem and will fit a logistic regression model in R, and only look into three way interactions to compare all the combinations:
# simulate data
income = sample(c("low", "middle", "high"), 1000, replace = T)
race = sample(c("white", "non-white"), 1000, replace = T)
education = sample(c("low", "middle", "high"), 1000, replace = T)
y = rbinom(1000,1,c(0.1, 0.5, 0.9, 0.8, 0.4))

df = data.frame(y=y, income = income, race=race, education = education)

fit = glm(y ~ (income+race+education)^3 - (income+race+education)^2 - 1, data = df, family = binomial(link = 'logit'))
summary(fit)

You can visually compare the effects using GGally package like this:
GGally::ggcoef_model(fit)   


